Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 2, April 2022
Page(s) 99 - 103
DOI https://doi.org/10.1051/wujns/2022272099
Published online 20 May 2022

© Wuhan University 2022

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

The goal of this paper is to study the relationship between the regularity and energy conservation of the nonhomogeneous incompressible ideal magnetohydrodynamics (MHD) equations, reading as:(1) (2) (3) (4)where ρ, u and B represent density, speed and the magnetic field, respectively. The permeability μ is a constant, and the pressure P is unknown. We consider this system in the three-dimensional periodic domain T 3.

When B=0, system (1)-(4) become the well- known nonhomogeneous incompressible Euler equations. Onsager[1] asserted that the velocity of Hölder continuity with exponential δ>1/3 guarantees energy conservation, and many researchers also studied the energy conservation of the Euler equations, which can be referred to Refs.[2-4] etc.

For the homogeneous ideal MHD equations, Caflisch, Klapper and Steele[5] proved the conservation of energy in the periodic domain by extending the results of the Euler equations to the ideal MHD equations. Then Kang and Lee[6] proved the conservation of energy and cross-helicity of the ideal MHD equations. Subsequently, Yu[7] improved the previous results by using the special structure of the nonlinear term in the ideal MHD equations.

For the nonhomogeneous MHD equations, some results have also been obtained recently, which could be referred to Refs.[8-13] etc. Among these documents, Bie et al [8] studied the compressible MHD equations, and they gave two sufficient conditions on the regularity of solutions to ensure energy conservation. Wu et al [9] considered the incompressible MHD equations, and they proved that the regularity of the solution is sufficient to guarantee the balance of the total energy in the Besov space. However the magnetic field in Ref. [9] needs to satisfya natural question then is whether this condition is required or not.

Inspired by Refs. [3, 4, 8] and [9], we study the energy conservation of nonhomogeneous incompressible ideal MHD system (1)-(4) in the three-dimensional periodic domain T 3. Our strategy relies on commutator estimates similar to those employed by Constantin et al [4]. In order to prove the energy conservation of system (1)-(4), we first mollify the equation (1) and then select the test function ϕ to obtain an equation that approximates the energy equation. For the treatment of the nonlinear term in this equation, we use the condition to transfer the derivative to the test function ϕ by partial integral formula, which makes the regularity assumption about curlB in Ref.[9] unnecessary.

We briefly recall the definitions of Lebesgue space and Besov space , where or . Lebesgue whereAnd , wherehere .

The main result of this paper is stated as follows.

Theorem 1   Let be a weak solution to system (1)-(4). Assumefor some and such that(5)Then the energy is locally conserved in the sense of distribution, that is,(6)

Remark 1   If B=0, this system reduces to the nonhomogeneous incompressible Euler equations, and our result could recover the one for the incompressible Euler equations[3].

1 Preliminaries

In this section we introduce some properties of Besov space . Let for d=3 or d=4 (according to the choice of Ω) be a standard mollifying kernel and setwith the notation . For any function ω, is well-defined on

Referring to Ref. [1], we have the following facts about functions in the Besov spaces:(7)and(8)

Moreover, is an algebra, that is the product of two functions in this space is again contained in the space.

2 Proof of Theorem 1

By smoothing (1) in space and time, we obtain(9)Taking , multiplying (9) by and integrating on , we get(10)here we take small enough so that . We can rewrite (10), using appropriate commutators, as(11)where

In order to prove our theorem, the first step is to show that the terms in the left side of (11) converge to the ones of the equation (6). For the first three terms of the left side of (11), we refer to the proof of Ref. [3]( Theorem 3.1), and get(12) (13) (14)And for the fourth term, we first introduce the following equality about curl,By smoothing (3) and multiplying it by , we obtainThen(15)

Thus, combining (11), (12), (13), (14) and (15), we find(16)To prove our result, it suffices to show , , as . For the treatment of and , we refer to the proof of Theorem 3.1 of Ref. [3], and get(17)and(18)here we request . In fact when , we know andwhich yields that . As for , in view of the equalitywe have(19)Here can be estimated asBecause the estimates of I1 and I2 are the same, we only calculate I1. Firstly we observe that

By using Hölder inequality and Fubini theorem, we estimate I1 as(20)and(21)The estimate for is similar to that of I1. Then(22)Collecting all the above estimates and putting them into (16), we obtainwhich completes the proof of Theorem 1..

References

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